1. Introduction

Micro-light emitting diodes ($\mathrm {\mu }$-LEDs) have attracted increasing attention in recent years and are expected to be used for next generation self-emissive displays. The outstanding electrical and optical properties of $\mathrm {\mu }$-LED displays surpass those of current display technologies such as LCDs or OLEDs when it comes to contrast ratio, brightness, resolution, response time and life time. [1–3] Furthermore, $\mathrm {\mu }$-LED arrays with new dimensions of pixel densities up to 8,500 pixels per inch have already been demonstrated [4]. Such properties make $\mathrm {\mu }$-LEDs very interesting for a plethora of visualization-related applications. Therefore, great efforts are spent in the development of $\mathrm {\mu }$-LED arrays consisting of tiny pixels in the single micrometer size regime.

One major drawback of $\mathrm {\mu }$-LEDs is high losses in internal quantum efficiency (IQE) arising with the miniaturization due to non-radiative recombination channels at pixel sidewalls. Especially with shrinking pixel sizes down to the range of one micrometer, sidewall recombinations are increasingly important due to the high surface-to-volume ratio. This effect is especially critical for red-emitting $\mathrm {\mu }$-LEDs based on AlGaInP material system, whereas it seems to be less relevant in InGaN based $\mathrm {\mu }$-LEDs. Significant reduction of this effect in AlGaInP or finding a red-emitting material based on nitride semiconductors are currently the main challenges in terms of the IQE that still need to be overcome and are necessary towards the realization of full-color $\mathrm {\mu }$-LED displays. [5]

In addition to the drastic reduction of the IQE at reduced pixel sizes, the light extraction also is expected to change significantly once the pixel size reaches the wavelength of the emitted light where wave optics must be considered. A systematic investigation of these optical changes is still missing and basic relations between pixel size, pixel morphology and emission wavelength are not well studied yet, even though there is a general agreement in literature that the LEE of $\mathrm {\mu }$-LEDs increases with decreasing pixel size [6–9].

Finite-difference time-domain (FDTD) simulations from recent publications were performed mainly under use of one single dipole source in the pixel center for pixel sizes from 5 µm to 20 µm [10–13]. This assumption may give a first estimate of the LEE trends, but it cannot realistically describe the optical properties of the pixel. In general, light is generated in the complete active area region of a $\mathrm {\mu }$-LED and not only in the pixel center. In this work, we therefore use several dipole sources distributed among the complete active area region from pixel center to pixel edge. The results of this more comprehensive method deviate significantly compared to using a single dipole source in the pixel center (Supplement 1, Section 6). In other works ray tracing methods were used for $\mathrm {\mu }$-LEDs down to 2 µm [7,14], which also might show qualitative trends, but it does not allow to study wave effects inside the pixel as the pixel dimensions become smaller.

Although power efficiency and hence a high LEE is essential, a $\mathrm {\mu }$-LED display also needs to fulfill other optical properties for real world applications. For instance, far field properties and color stability play an important role at system level. This study focuses on the impact of the pixel design and of the emission wavelength on the optical properties including both the absolute LEE and far field properties using FDTD simulations.

2. Model and methods

2.1 Pixel design

For our study we use a simplified model of a single $\mathrm {\mu }$-LED pixel within the InGaN material system. A chip architecture is used where the light is extracted via the n-side in the final device. Therefore, a reflective mirror is deposited on the p-side and then bonded on a Si-carrier substrate. In the thin film flip-chip concept the growth substrate is then removed and subsequently the epitaxial layers on the n-side, such as buffer layers can be removed or thinned. Figure 1 shows the schematic cross section of a circular pixel with a certain pixel sidewall angle $\alpha$ and pixel diameter $d$, where $d$ refers to the span of the active area region. The rotational symmetry of the circular pixel is used to perform two-dimensional simulations of the cross-section and then calculate the light extraction characteristics for the three-dimensional case. From a practical point of view round structures might be easier to fabricate compared to rectangular shape due to photolithography resolution and other process limitations such as wet chemical etching. This is especially the case for pixels approaching one micrometer or even smaller.

 

Fig. 1. Schematic model of the pixel design used within the thin film flip-chip architecture where light is extracted via the n-side with removed growth substrate. The impact of the pixel sidewall angle $\alpha$ and the pixel diameter $d$ on the light extraction is in the focus of this work for different emission wavelengths.

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The pixel height in our design amounts to 600 nm for all pixel diameters which is smaller than all pixel diameters and an average value compared with other experimental and theoretical studies [7,10,12,15]. A 50 nm thick Indium tin oxide (ITO) film covers the p-side of the etched pixel. The 150 nm thick dielectric layer out of $\textrm{SiO}_{2}$ isolates the InGaN semiconductor from the silver mirror. The remaining n-GaN thickness after removal of the epitaxial buffer layers amounts to 1.5 µm and has a smooth interface to air. For n-GaN thicknesses between 1.0 $\mathrm {\mu }$m and 4.0 $\mathrm {\mu }$m there is no significant change in the obtained results in terms of both LEE and far field shape (Supplement 1, Section 7). We intentionally omit a p-contacting scheme of the individual pixels and furthermore various epitaxial layers commonly used in commercial LEDs are not considered, since we want to focus on fundamental optical properties purely caused by the pixel design and emission wavelength. This simplification will not affect the overall trends and findings of this study. [16]

Three emission wavelengths representing the colors of an RGB display are studied within this nitride-based $\mathrm {\mu }$-LED pixel. So far the InGaN material system is typically used for blue and green LEDs, but several approaches to achieve red-emitting InGaN have recently shown promise, such as using selective area growth or growth on specially prepared growth substrates [15,17–19]. Thus, the optical characteristics of $\mathrm {\mu }$-LEDs with InGaN QWs emitting at 460 nm, 530 nm and also 615 nm are studied, with each emission spectrum having a FWHM of 30 nm. Variations are expected due to wavelength dependent material parameters as well as interference and mode effects at smallest pixel sizes. The comparison of these pixels emitting at different wavelengths is realized by individual pixel models where the respective InGaN QW is ideally placed close to the $5/4 \lambda$ optical path length with respect to the reflective mirror for each emission wavelength. The models used are shown in Fig. S1 (Supplement 1) and all parameter values and dimensions are listed in Table S1 for each color, respectively.

2.2 Implementation

Two-dimensional FDTD simulations were performed with the commercially available program Ansys Lumerical. Here, an electric dipole source is used for the generation of an electromagnetic wave within a model of the pixel environment and Maxwell’s equations are solved numerically on a discrete spatial mesh grid for different time steps. The output quantities are the electric field $\vec {E}$ and magnetic field $\vec {H}$ at each point of the simulation domain. From these fields several optical properties can be extracted.

The pixel design described in section 2.1 is implemented within a parameterized model with a mesh size of 6 nm (see Fig. S1, Supplement 1). The material permittivity parameters used were taken from literature [20–22] (see Fig. S2, Supplement 1). In addition to the dielectric constants used an overview and description of all other parameters used is given in Table S1. The minimum simulation time of 150 fs was chosen along with other parameters to guarantee a converged simulation result. A perfectly matched layer is used as the boundary condition of the simulation domain. Dipole sources are distributed across the active area with a spacing $\Delta r = {50}\;\textrm{nm}$ at positions $r_i = \left \{ 0,\Delta r,2\Delta r, \dots, d/2 - \Delta r\right \}$, each with four dipole orientations $\varphi _j, \theta _k = \{0,\pi /2\}$. Here, $\theta _k = 0$ corresponds to the out-of-plane and $\theta _k = \pi /2$ corresponds to the in-plane electric dipole source. $\varphi _j$ is the angle between the electric dipole vector and the positive $y$-axis. The material parameters of the active region are assumed to be the same as the surrounding GaN material. Furthermore, p-GaN and n-GaN are not distinguished in this study. An individual simulation was performed for each dipole position and orientation, as previously done by Ryu et al. [6,23].

Figure 2 shows the two-dimensional electric field distribution resulting from a linear superposition of several dipole simulations, exemplary for a pixel with $d= {1}\;\mathrm{\mu}\textrm{m}$ and $\alpha = {45}^{\circ}$. In this case the upper graph only shows TE dipoles pointing in $y$-direction ($\varphi = 0$) and the lower graph shows TE dipoles pointing in $x$-direction ($\varphi = \pi /2$). The reflection of the emitted light for $\varphi = 0$ orientated dipoles is dominated by the horizontal mirror below the ITO. In contrast, the reflection for $\varphi = 90$ orientated dipoles is dominated by the mirrors on the pixel sidewalls. The electric field distribution of the TM dipoles for this pixel is shown in Fig. S3 (Supplement 1) and the reflection here is divided among the horizontal mirror and the mirror on the pixel sidewalls.

 

Fig. 2. Two-dimensional electric field distribution for a $\mathrm {\mu }$-LED pixel with $d= {1}\;\mathrm{\mu}\textrm{m}$ and $\alpha = {45}^{\circ}$ obtained by a linear superposition of several individual FDTD simulations for different dipoles distributed across the active area. In the upper graph only dipoles with orientations $\theta = \pi /2, \varphi =0$ (in-plane dipole vector pointing in $y$-direction) are shown, whereas the lower graph has dipoles with orientation $\theta = \pi /2, \varphi =\pi /2$ (in-plane dipole vector pointing in $x$-direction). Both graphs share the same color intensity map. Black lines indicate interfaces between different materials.

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In addition, electric field distributions for various pixel designs can be found in Figs. S5 and S6 (Supplement 1) which allow to get some insights how the pixel size or rather the pixel sidewall angle $\alpha$ act on the light extraction. These visualizations indicate also why it is not sufficient to use a single dipole in the pixel center and further also different dipole orientations need to be considered to evaluate the optical properties of a $\mathrm {\mu }$-LED.

2.3 Data processing

The detected light power of the monitor is calculated by a scalar product of the Poynting vector $\vec {S} = \vec {E}\times \vec {H}$ and the surface normal $\vec {n}$ for all mesh points within the detection plane over a certain time. In this work, the wavelength-dependent outcoupling efficiency $\eta (\lambda )$ is the ratio between the detected light power at the monitor in air and dipole source power. It is calculated according to Eq. (1).

Here, the source power is calculated by using a box of four monitors surrounding the dipole source. For a specific dipole source with emission spectrum $S(\lambda )$ the light outcoupling efficiency of one dipole at position $r_i$ and orientation $\varphi _j, \theta _k$ is given by Eq. (2).

In our case the hypothetical emission spectrum $S(\lambda )$ follows a Gaussian distribution with respective center wavelengths 460 nm, 530 nm and 615 nm and a finite emission bandwidth with $\text {FWHM}= {30}\;\textrm{nm}$.

The total LEE of a pixel is calculated according to Eq. (3) where $\eta (r_i, \varphi _j, \theta _k)$ is the light extraction efficiency of one single dipole at position $r_i$ and orientation $\varphi _j, \theta _k$. $w_1(r)$, $w_2(\varphi, \theta )$ and $w_3(r)$ are weighting factors.

The first weighting factor $w_1(r_i)$ takes into account the geometric arrangement for three-dimensional pixels with dipoles distributed over the entire circular active area.

The second factor $w_2(\varphi _j, \theta _k)$ considers a different weighting for differently orientated electric dipoles. We assume a simplified weighting $w_2(\varphi _j, \theta _k)=w_2(\theta _k)$ depending solely on the mode of the emitted waves:

Other position dependent contributions acting on the dipole strength are reflected in the weighting factor $w_3(r)$. This can, e.g., account for a different weighting of the dipoles in the pixel center compared to the pixel edge due to local IQE differences [24] or the result of an inhomogeneous current distribution within the pixel [12] leading to different dipole powers.

Far field patterns are calculated from individual simulations out of the electric field distribution of the monitor in air. Afterwards a superposition of these far fields with weighting in the same manner as for the LEE was performed to obtain the far field of one pixel.

To determine the actual TE/TM ratio, one would require further band structure simulations or experimental data, which is beyond the scope of this work. Therefore, we assume a 1:1 ratio for the first part ($w_2(\theta ) = 1$) and discuss a change in section 3.3. Furthermore, for the sake of simplicity, a spatially uniform dipole strength across the whole active area is initially assumed ($w_3(r)=1$), but as well modified and discussed in section 3.3.

3. Results and discussion

3.1 Light extraction efficiency

A first investigation was done for blue emitting pixels with sizes from 1 µm to 20 µm and two different sidewall angles $\alpha = 0^\circ$, $30^\circ$. The obtained LEE data is shown in Fig. S4 (Supplement 1) and shows a strict increase with decreasing pixel diameter $d$ as previously predicted by other simulations and experimentally reported [6–9]. We found a LEE improvement of 45% for a 1 $\mathrm {\mu }$m blue emitting pixel compared to a 20 $\mathrm {\mu }$m pixel at $\alpha = {30}^{\circ}$. From 5 $\mathrm {\mu }$m to 1 $\mathrm {\mu }$m the improvement amounts to 25% for the same sidewall angle. For vertical pixel sidewalls, this improvement is 21% and 13% for a 1 $\mathrm {\mu }$m pixel compared to a 20 $\mathrm {\mu }$m and 5 $\mathrm {\mu }$m pixel respectively. It is apparent that the pixel sidewall angle $\alpha$ has a higher impact on small blue pixels ($d= {1.0}\;\mathrm{\mu}\textrm{m}$) allowing to improve the LEE by a factor of 1.4 at $\alpha =30^\circ$ compared to vertical sidewalls. For pixels with $d= {20}\;\mathrm{\mu}\textrm{m}$ the impact of the sidewall angle becomes less relevant and shows a factor 1.07 enhancement between vertical sidewalls and $\alpha = 30^\circ$ corresponding to an absolute improvement of 0.03. This offset of 0.03 does not depend on the pixel size down to a pixel diameter of $d= {5}\;\mathrm{\mu}\textrm{m}$, where we observe an increasing impact of the sidewall angle $\alpha$ and pixel diameter $d$.

Therefore, in the following studies the focus lies on pixels with diameters $d$ in the range from 0.8 µm up to 5.0 µm and sidewall angles $\alpha$ in the range from 0° (vertical sidewalls) to 60°.

Figure 3 shows the absolute LEE depending on the parameters $d$ and $\alpha$ at different emission wavelengths. The upper row of color maps represents here the total LEE covering all light extracted to air (in the full hemisphere from a $- 90^\circ$ to a $+ 90^\circ$ far field angle) while the lower row focuses on the light extracted within a $\pm 15^\circ$ solid angle. For cases when the emitted light is coupled into optics with small acceptance angle this value is relevant for the final efficiency of the optical system. The data shows a clear increase in LEE with decreasing pixel diameter $d$ for all three colors. Further it shows that the pixel sidewall angle $\alpha$ is a powerful design parameter when it comes to LEE optimization, especially for smallest pixels in the single micrometer size regime. We find that the optimum $\alpha$ is almost independent of the pixel size $d$, but changes with the emission wavelength. Thus, the optimum sidewall angle $\alpha$ is 35° for blue or 40° for green and red emitting pixels. The respective maximum LEE at these angles amounts to 0.57, 0.54 or 0.57 for blue, green or red emitting pixels each for the smallest 0.8 $\mathrm {\mu }$m pixels. At a pixel size of $d= {1.0}\;\mathrm{\mu}\textrm{m}$ and a fixed sidewall angle of $\alpha =30^\circ$ the LEE drops from 0.52 to 0.47 considering blue and red emitting pixels. This implies that different pixel designs should be considered for different emission wavelengths, which have an essential impact on the LEE.

 

Fig. 3. LEEs for different $\mathrm {\mu }$-LEDs with pixel diameter $d$ and pixel sidewall angle $\alpha$ for each a 460 nm-blue, 530 nm-green and 615 nm-red emitting QW with a finite FWHM of 30 nm. The upper row of plots shows the total LEE including all light extracted at the n-side (from a $- 90^\circ$ to a $+ 90^\circ$ far field angle) while the second row focuses on the light extracted within a $\pm 15^\circ$ far field angle. All three plots of each row share the same color bar plotted at the right hand side.

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The calculated LEE when considering only the emitted light within a far field angle of $- 15^\circ$ to $+ 15^\circ$ is 0.19, which is 35% of the total LEE for blue pixels (0.16 or 33% and 0.15 or 31% for green and red respectively). Besides these absolute differences, the overall LEE tendencies as a function of $\alpha$, $d$ and wavelength observed for the case of hemisphere detection are not anymore valid. A monotone increase in LEE with decreasing pixel size is not observed. The emission wavelength significantly changes the overall trend in the representation of $d$ versus $\alpha$, showing a global maximum for blue pixels and several local maxima for green and red pixels. This is a first indicator that the far field plays an important role for pixels between 0.8 $\mathrm {\mu }$m and 2.0 $\mathrm {\mu }$m as the wavelength becomes comparable to the pixel size. The deviating trends for green and red emitting pixels smaller than 2.0 $\mathrm {\mu }$m indicate that wave optical effects within the pixel become more visible as the wavelength approaches the pixel dimension. Consequently, the far field changes sensitively even for small changes in $d$ and $\alpha$ and thus results in significant LEE differences within $\pm 15^\circ$. For blue emitting $\mathrm {\mu }$-LEDs there is a difference between the global maximum found at $d= {1.1}\;\mathrm{\mu}\textrm{m}$ and $\alpha =35^\circ$ at small acceptance angles and the hemisphere maximum that is at $d= {0.8}\;\mathrm{\mu}\textrm{m}$ and $\alpha =35^\circ$. This is explained by the fact that the far field intensity distribution is more centered around the $0^\circ$ far field emission. For green and red emitting pixels the trend changes and shows several local maxima. The LEE color maps for green and red emitting pixels show a poorly contoured maximum at $d= {1.3}\;\mathrm{\mu}\textrm{m}$, $\alpha =35^\circ$ and $d= {0.8}\;\mathrm{\mu}\textrm{m}$, $\alpha =40^\circ$, respectively.

3.2 Far field characteristics

In the following, we analyze in more detail the far field properties, which play a relevant role based on the respective display application. The first row of plots in Fig. 4 shows the far field intensity distributions of three blue emitting pixels with diameters $d$ of 1.0 µm, 2.0 µm and 5.0 µm. The plotted curves in different colors represent pixels with different sidewall angles $\alpha$. The far field shape undergoes a substantial transition when shrinking the lateral pixel dimension from 5.0 µm down to 1.0 µm. While emission patterns for the larger pixel follow Lambert’s emission law and have less dependence on the pixel sidewall angle $\alpha$, the 1.0 µm pixels have a far field shape with several lobe-like features depending on the pixel sidewall angle $\alpha$. It can be seen that both the far field shape and the absolute intensity are increasingly influenced by $\alpha$ as the pixels dimension is approaching one micrometer. The change in shape caused by different pixel sidewall angles $\alpha$ is less pronounced for the 2.0 µm pixel, but still the absolute intensity shows a high variation. In general, the same trends can be observed for green and red emitting pixels as well as other pixel sizes and sidewall angles (see Figs. S7, S8, and S9 in Supplement 1). These characteristic far field shapes result from wave optical effects inside the pixel and are not easily predictable or comprehensible. The very specific lobe-like features result from a nonlinear combination of intrinsic events.

The second row of plots in Fig. 4 shows calculated center wavelengths $\lambda _\text {center}$ from different spectra of the outcoupled light as a function of the far field angle for blue, green and red emitting pixels with a sidewall angle $\alpha$ of $30^\circ$. All pixels show mean center wavelengths $\lambda _\text {center}$ deviating from the respective emission wavelength of 460 nm, 530 nm or 615 nm by a few single nanometers. The center wavelength as a function of the far field angle shows substantially different behavior for blue, green and red emitting pixels. For pixel sizes of 2.0 $\mathrm {\mu }$m and smaller, the pixel sidewall angle $\alpha$ also has a large effect on the evolution of $\lambda _\text {center}$ (see Figs. S10, S11, and S12 in Supplement 1). The obtained data reveals that the center wavelength $\lambda _\text {center}$ shifts over a range of 2 nm for a 1.0 $\mathrm {\mu }$m blue pixel, while $\lambda _\text {center}$ shifts over 6 nm for a 1.0 $\mathrm {\mu }$m red pixel. One possible reason that red is more sensitive is because the wavelength is closer to the pixel dimension.

 

Fig. 4. Far field properties for different pixel designs: The upper row shows far field patterns for different blue emitting pixels. The far field undergoes a clear transition with the down-shrinking from 5.0 $\mathrm {\mu }$m large pixels down to 1.0 $\mathrm {\mu }$m pixels. The lower row shows the center wavelength of the outcoupled light spectrum depending on the far field angle for the three different RGB colors. The pixel sidewall angle amounts here $30^\circ$. This trend shows that small red emitting pixels are more sensitive with respect to these far field properties.

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We showed that the far field characteristics undergo a substantial transition for smallest $\mathrm {\mu }$-LEDs. Therefore, in the design process of $\mathrm {\mu }$-LED arrays, it is important to consider that the far field deviates strongly from Lambert’s emission law when the pixel size is reduced below 2.0 $\mathrm {\mu }$m. Furthermore, we found a major difference for blue, green and red emitting pixels, not only when analyzing the far field shape, but also when evaluating the center wavelength as a function of the far field angle. The wavelength shift of 2 nm for blue and 6 nm for red emitting pixels can be seen as a color change when the pixel is viewed from the side. This color change is also distinguishable by the human eye which is sensitive to a wavelength difference of 2 nm for blue and green light and even more sensitive down to 1 nm for longer wavelengths in the spectral red range [25]. Depending on the display application this property can be of technological relevance and needs to be considered within the design process.

3.3 Discussion on impact of model parameters

In this section we study the impact of three parameters concerning the dipoles: The individual dipole contribution depending on its radial position, the TE/TM emission ratio and the dipole emission bandwidth. In the previous sections we have assumed an ideal equal weighting of all dipoles within the active area and considered only the geometric distribution with $w_1(r)$ in Eq. (4). For a real $\mathrm {\mu }$-LED it is unlikely to happen that each dipole has the same emission power or in other words it is likely that the light emission power within the QW depends on the radial position within the pixel. Reasons for this could be a current distribution profile within the pixel as already simulated by Hang et al. [12] or other physical phenomena like non-radiative recombinations acting on the local IQE [24]. Basically, both scenarios can lead to dipoles emitting with lower intensity towards the edge of the pixel than in the center of the pixel. This can be taken into account by an additional position-dependent weighting factor $w_3(r)$. We introduce two empirical weighting functions $w_3(r)$ represented by a Gaussian with a FWHM of $d$ or $d/2$ and recalculate the LEE according to Eq. (3). The used Gaussian weighting functions can be seen on the top side of Fig. 5 as a function of the radial distance from the pixel center and below the absolute change in LEE of blue $\mathrm {\mu }$-LEDs compared to the constant $w_3(r)=1$ as used beforehand in Sections 3.1 and 3.2. The maximum change in LEE for both weighting functions occurs at a pixel size of 0.8 $\mathrm {\mu }$m and sidewall angle $\alpha =60^\circ$. It shows an absolute improvement of $+0.01$ or $+0.05$ for a Gaussian like $w_3(r)$ with a FWHM of $d$ or $d/2$ respectively compared to the constant weighting $w_3(r)=1$. For a blue 1.0 $\mathrm {\mu }$m pixel with a sidewall angle of $30^\circ$ the absolute change in LEE only amounts $+0.004$ or $+0.014$. Green pixels show a similar trend as blue pixels with LEE improvement for most pixels, while a distinct group of red emitting pixels even show a loss in LEE (see Fig. S13, Supplement 1). Moreover, the far field shapes change noticeably with a different weighting and the intensity maximum gets closer to the $0^\circ$ far field angle for the 1.0 µm pixel (see Fig. S14, Supplement 1). In conclusion the additional weighting factor $w_3(r)$ has a limited impact on the absolute hemisphere LEE, but causes an observable change in the far field pattern.

 

Fig. 5. Impact of an additional position dependent dipole weighting factor $w_3(r)$ on the absolute LEE for blue emitting pixels. The upper graph shows the used weighting functions $w_3(r)$. In the middle and lower graph the absolute differences in LEE with respect to the case with $w_3(r)=\text {const.}$ are shown. The two investigated weighting functions are described by a Gaussian with either a FWHM of $d$ or $d/2$, where $d$ is the respective pixel diameter.

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An equal contribution of both TE and TM modes was assumed for the results shown in Sections 3.1 and 3.2. Now we set the TE fraction of Eq. (5) to twice the TM fraction to examine the change in LEE with a different TE/TM ratio. Figure S15 (Supplement 1) shows the change in LEE compared to the previously used equal weighting. For all emission wavelengths and all pixel designs, we find a maximum deviation of -0.02 in the absolute LEE, which means that the simulation results and especially the general trends are meaningful even if we do not know the exact TE/TM ratio. Furthermore, the dipole modes have only minor impact on the different far field shapes and mainly differ by a scalar factor (far field patterns can be found in Fig. S16, Supplement 1).

Finally, the impact of the emission bandwidth was investigated by using a FWHM of 45 nm or rather 60 nm instead of the assumed 30 nm for all three emission spectra. Here, the general trends and absolute LEEs do not change significantly with higher bandwidths (see Fig. S17, Supplement 1). The absolute difference in LEE is at most $-0.02$ for almost all pixel designs and colors. However, the far field patterns are more affected by a higher emission bandwidth and especially for the 1.0 µm pixel the intensity distribution across the far field angle is more smeared out (Figs. S18, S19, and S20 in Supplement 1).

With these investigations, we have proven that the derived LEE results and far field characteristics can generally be trusted even in the case when some parameters such as the TE/TM ratio or the emission bandwidth are a priori unknown. Furthermore, an additional weighting factor, with which dipoles at the pixel edge were half weighted compared to dipoles in the pixel center, showed minor changes in hemisphere LEE. Obviously, there are other assumptions and unknown parameters in our simulation model that we have not explored in detail, but which could possibly affect the results. This comprises surface roughness at material interfaces, rounded edges, a p-contacting scheme, material parameters, additional Epi layers including different doping concentrations and asymmetry of the pixel.

4. Conclusion

By using several individual FDTD simulations with dipoles distributed in 50 nm steps over the active area of a single pixel and considering different dipole orientations, we systematically investigated fundamental optical properties of $\mathrm {\mu }$-LEDs in the single micrometer size regime. We revealed general LEE trends for various pixels of sizes in the range from 0.8 µm up to 5.0 µm and sidewall angles $\alpha$ in the range from 0° to 60°. In contrast to prior works a special focus of this study lied on the investigation of different emission wavelengths and far field characteristics.

The simulations yield that the optimum pixel sidewall angle $\alpha$ is 35° for blue or 40° for green and red emitting pixels independent of the pixel size. Especially, for smallest pixels approaching one micrometer the sidewall angle is a powerful design parameter to improve the LEE. We addressed the impact of emission wavelength on the LEE and far field properties and found that mainly the far field is affected and shows substantial differences. Therefore, one must consider different pixel designs for the three required colors that form an RGB display. By changing different weighting factors in the data processing the effect of assumptions we made was investigated and discussed. It demonstrated that our results and the general trends can be considered with confidence.

After exploring the fundamental optical properties caused by the pure pixel geometry and emission wavelength, future studies could address the influence of a p-contacting scheme or the effects of different epi-layers for realistic pixels. In addition, it is important to further investigate the role of asymmetry in the fabrication of $\mathrm {\mu }$-LEDs. Since the control options of the far field properties are rather limited by the pixel sidewall angle adjustment, it is reasonable to further consider light extraction structures in combination with angled sidewalls.